iPhone 5 won't turn on after two battery replacements, is it dead?

Hello I was wondering if anyone could help out, I have an iPhone 5 which I purchased in Dec 2012 that stopped working just after my warranty expired & about a week after I upgraded to ios 7...

Basically it wouldn't turn on or charge, I bought new chargers tried the holding button thing but nothing, not even the empty battery screen. So I took it to a phone repair shop who said they'd need to replace the battery and screen (they insisted my screen was broke) but whatever I paid and all was good. My phone was working but after one day the same thing happened. Again it wouldn't charge or turn on.

I went back to the shop but they wouldn't help so I bought a battery online and done it myself this time. It worked! I then completely wiped my phone and reinstalled ios but again after a couple of days my battery was dead and my iPhone wouldn't turn on or charge.

That's two batteries dead (three including Apples!), is my phone dead? Is there any hope? I know that I could probably just purchase a bunch of batteries and keep replacing them each time they but you know. Just to note I'm not positive on this but I think all of the batteries died/wouldn't charge after they completely ran out but were charging fine whilst they were on.

Any idea? I've not really got money for another phone and I wasn't planning on an upgrade until December so I'd desperately like my phone to last until then. Thanks.